Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ineccnvmo Structured version   Visualization version   GIF version

Theorem ineccnvmo 34464
Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
Assertion
Ref Expression
ineccnvmo (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo
StepHypRef Expression
1 relcnv 5644 . . 3 Rel 𝐹
2 id 22 . . . 4 (𝑦 = 𝑧𝑦 = 𝑧)
32inecmo 34462 . . 3 (Rel 𝐹 → (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥))
41, 3ax-mp 5 . 2 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥)
5 brcnvg 5441 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝐹𝑥𝑥𝐹𝑦))
65el2v 34329 . . . 4 (𝑦𝐹𝑥𝑥𝐹𝑦)
76rmobii 3282 . . 3 (∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∃*𝑦𝐵 𝑥𝐹𝑦)
87albii 1895 . 2 (∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
94, 8bitri 264 1 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 834  wal 1629   = wceq 1631  wral 3061  ∃*wrmo 3064  Vcvv 3351  cin 3722  c0 4063   class class class wbr 4786  ccnv 5248  Rel wrel 5254  [cec 7894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7898
This theorem is referenced by:  ineccnvmo2  34467
  Copyright terms: Public domain W3C validator